Slides by John Loucks St. Edward’s University

1. Slides by John Loucks St. Edward’s University

2. Chapter 2, Part BDescriptive Statistics:Tabular and Graphical Presentations • Exploratory Data Analysis: Stem-and-Leaf Display • Crosstabulationand Scatter Diagram

3. Exploratory Data Analysis • The techniques of exploratory data analysis consist of • simple arithmetic and easy-to-draw pictures that can • be used to summarize data quickly. • One such technique is the stem-and-leaf display.

4. Stem-and-Leaf Display • A stem-and-leaf display shows both the rank order • and shape of the distribution of the data. • It is similar to a histogram on its side, but it has the • advantage of showing the actual data values. • The first digits of each data item are arranged to the • left of a vertical line. • To the right of the vertical line we record the last • digit for each item in rank order. • Each line in the display is referred to as a stem. • Each digit on a stem is a leaf.

5. Example: Hudson Auto Repair The manager of Hudson Auto would like to gain a better understanding of the cost of parts used in the engine tune-ups performed in the shop. She examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar, are listed on the next slide.

6. Stem-and-Leaf Display • Example: Hudson Auto Repair Sample of Parts Cost ($) for 50 Tune-ups

7. Stem-and-Leaf Display • Example: Hudson Auto Repair 5 6 7 8 9 10 2 7 2 2 2 2 5 6 7 8 8 8 9 9 9 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9 0 0 2 3 5 8 9 1 3 7 7 7 8 9 1 4 5 5 9 a stem a leaf

8. Stretched Stem-and-Leaf Display • If we believe the original stem-and-leaf display has • condensed the data too much, we can stretch the • displayvertically by using two stems for each • leading digit(s). • Whenever a stem value is stated twice, the first value • corresponds to leaf values of 0 - 4, and the second • value corresponds to leaf values of 5 - 9.

9. Stretched Stem-and-Leaf Display • Example: Hudson Auto Repair 5 5 6 6 7 7 8 8 9 9 10 10 2 7 2 2 2 2 5 6 7 8 8 8 9 9 9 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9 0 0 2 3 5 8 9 1 3 7 7 7 8 9 1 4 5 5 9

10. Stem-and-Leaf Display • Leaf Units • A single digit is used to define each leaf. • In the preceding example, the leaf unit was 1. • Leaf units may be 100, 10, 1, 0.1, and so on. • Where the leaf unit is not shown, it is assumed • to equal 1. • The leaf unit indicates how to multiply the stem- • and-leaf numbers in order to approximate the • original data.

11. Example: Leaf Unit = 0.1 If we have data with values such as 8.6 11.7 9.4 9.1 10.2 11.0 8.8 a stem-and-leaf display of these data will be Leaf Unit = 0.1 8 9 10 11 6 8 1 4 2 0 7

12. Example: Leaf Unit = 10 If we have data with values such as 1806 1717 1974 1791 1682 1910 1838 a stem-and-leaf display of these data will be Leaf Unit = 10 16 17 18 19 8 The 82 in 1682 is rounded down to 80 and is represented as an 8. 1 9 0 3 1 7

13. Crosstabulations and Scatter Diagrams • Thus far we have focused on methods that are used • to summarize the data for one variable at a time. • Often a manager is interested in tabular and • graphical methods that will help understand the • relationship between two variables. • Crosstabulation and a scatter diagram are two • methods for summarizing the data for two variables • simultaneously.

14. Crosstabulation • A crosstabulation is a tabular summary of data for two variables. • Crosstabulation can be used when: • one variable is qualitative and the other is • quantitative, • both variables are qualitative, or • both variables are quantitative. • The left and top margin labels define the classes for • the two variables.

15. Crosstabulation • Example: Finger Lakes Homes The number of Finger Lakes homes sold for each style and price for the past two years is shown below. quantitative variable categorical variable Home Style Price Range Colonial Log Split A-Frame Total 18 6 19 12 55 45 < $200,000 > $200,000 12 14 16 3 30 20 35 15 Total 100

16. Crosstabulation • Example: Finger Lakes Homes Insights Gained from Preceding Crosstabulation • The greatest number of homes (19) in the sample • are a split-level style and priced at less than • $200,000. • Only three homes in the sample are an A-Frame • style and priced at $200,000 or more.

17. Crosstabulation Frequency distribution for the price range variable • Example: Finger Lakes Homes Home Style Price Range Colonial Log Split A-Frame Total 18 6 19 12 55 45 < $200,000 > $200,000 12 14 16 3 30 20 35 15 Total 100 Frequency distribution for the home style variable

18. Crosstabulation: Row or Column Percentages • Converting the entries in the table into row percentages or column percentages can provide additional insight about the relationship between the two variables.

19. Crosstabulation: Row Percentages • Example: Finger Lakes Homes Home Style Price Range Colonial Log Split A-Frame Total 32.73 10.91 34.55 21.82 100 100 < $200,000 > $200,000 26.67 31.11 35.56 6.67 Note: row totals are actually 100.01 due to rounding. (Colonial and > $200K)/(All > $200K) x 100 = (12/45) x 100

20. Crosstabulation: Column Percentages • Example: Finger Lakes Homes Home Style Price Range Colonial Log Split A-Frame 60.00 30.00 54.29 80.00 < $200,000 > $200,000 40.00 70.00 45.71 20.00 100 100 100 100 Total (Colonial and > $200K)/(All Colonial) x 100 = (12/30) x 100

21. Crosstabulation: Simpson’s Paradox • Data in two or more crosstabulations are often aggregated to produce a summary crosstabulation. • We must be careful in drawing conclusions about the • relationship between the two variables in the • aggregated crosstabulation. • In some cases the conclusions based upon an • aggregated crosstabulation can be completely • reversed if we look at the unaggregated data. The • reversal of conclusions based on aggregate and • unaggregated data is called Simpson’s paradox.

22. Scatter Diagram and Trendline • A scatter diagram is a graphical presentation of the • relationship between two quantitative variables. • One variable is shown on the horizontal axis and • the other variable is shown on the vertical axis. • The general pattern of the plotted points suggests • the overall relationship between the variables. • A trendline provides an approximation of the relationship.

23. Scatter Diagram • A Positive Relationship y x

24. Scatter Diagram • A Negative Relationship y x

25. Scatter Diagram • No Apparent Relationship y x

26. Scatter Diagram • Example: Panthers Football Team The Panthers football team is interested in investigating the relationship, if any, between interceptions made and points scored. x = Number of Interceptions y = Number of Points Scored 14 24 18 17 30 1 3 2 1 3

27. 35 30 25 20 15 10 5 0 1 0 2 3 4 Scatter Diagram y Number of Points Scored x Number of Interceptions

28. Example: Panthers Football Team • Insights Gained from the Preceding Scatter Diagram • The scatter diagram indicates a positive relationship • between the number of interceptions and the • number of points scored. • Higher points scored are associated with a higher • number of interceptions. • The relationship is not perfect; all plotted points in • the scatter diagram are not on a straight line.

29. Scatter Diagram and Trendline Scatter Diagram for the Panthers 35 30 25 20 Points Scored. Number of 15 10 5 0 0 1 2 3 4 Number of Interceptions

30. Tabular and Graphical Methods Data Categorical Data Quantitative Data Tabular Methods Graphical Methods Tabular Methods Graphical Methods • Bar Chart • Pie Chart • Frequency • Distribution • Rel. Freq. Dist. • Percent Freq. • Distribution • Crosstabulation • Dot Plot • Histogram • Ogive • Stem-and- • Leaf Display • Scatter • Diagram • Frequency • Distribution • Rel. Freq. Dist. • % Freq. Dist. • Cum. Freq. Dist. • Cum. Rel. Freq. • Distribution • Cum. % Freq. • Distribution • Crosstabulation

31. End of Chapter 2, Part B